All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
5
votes
0
answers
215
views
Unirationality over $\mathbb{Q}$
It is known that all smooth projective quartic hypersurfaces of suitably large dimension are unirational over $\overline{\mathbb{Q}}$. Are there any results regarding unirationality over $\mathbb{Q}$ ...
- 3,062
12
votes
2
answers
902
views
Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?
Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?
- 11.3k
2
votes
0
answers
230
views
Genus of $k(T)$ is $0$ without using Riemann-Roch
Let $F$ be a function field with one variable with total constant field $k$, and let $X$ be the set of all places of $F$. How do I show that the genus of $k(T)$ is $0$ without using Riemann-Roch? Is ...
- 29
3
votes
1
answer
382
views
Twisting by a multiplicative Character in Katz, Perversity and Exponential sums
Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided by ...
- 33
4
votes
1
answer
727
views
isogeny clases of CM abelian varieties
Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1_A)$ one gets a CM type of $K$, ...
- 43
2
votes
0
answers
242
views
Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$
when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$
I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that
$(x_{i},y_{i}),i=1,2,\cdots,N$ ...
- 4,280
15
votes
4
answers
1k
views
Number of $\mathbb F_p$ points constant mod $p$?
I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
- 27.9k
5
votes
2
answers
2k
views
Order of vanishing of an integer polynomial at a point
Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless $...
- 7,836
1
vote
2
answers
191
views
Pseudo-decision procedures for first order arithmetic
I was reading this paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.2911&rep=rep1&type=pdf in which the author describes an algorithm, based on Groebner basis, ...
- 19
13
votes
1
answer
974
views
Which degree does a motivic Galois representation show up in?
Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
- 149k
3
votes
0
answers
196
views
Unirationality and the Hasse principle
Is there an example of a quasiprojective variety $X$ defined over ℚ such that
$X$ is unirational over all finite fields, and
$X$ is unirational over $\mathbb{R}$, and
$X$ is not unirational over $\...
- 3,062
1
vote
0
answers
134
views
A question on exponential sums
Let $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ be a homogeneous polynomial, irreducible over $\mathbb{Q}$. Let $A$ be a positive constant, and let $B$ be a positive real number understood ...
- 26.9k
6
votes
0
answers
179
views
Geometric interpretation of Schmidt rank
For a form $f \in k[x_1, \cdots, x_n]$, where $k$ is a field of characteristic zero (or more specifically, a number field, and usually $\mathbb{Q}$). The Schmidt rank of $f$ (with respect to $k$), ...
- 26.9k
2
votes
0
answers
75
views
Question related to $h$-invariant of a form
Let $k$ be a field.
Given a form $f \in k[x_1, ..., x_n]$ of degree at least $2$, we define the
Schmidt rank, also known as the $h$-invariant, $h_k(f)$ to be the
least positive integer $h$ such that $...
- 579
3
votes
1
answer
196
views
Question about zeta function of function field in 1 variable over $\mathbb{F}_q$
From my previous question, I know that$$\zeta_X(s) = {{P(u)}\over{(1-u)(1-qu)}}$$for some polynomial $P(u)$ of degree $2g$, where
$X$ is the set of all places of $F$, a function field in one variable ...
- 75
4
votes
1
answer
601
views
Reference request, zeta function is rational function via Riemann-Roch?
I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...
- 75
4
votes
2
answers
742
views
Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)
I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.
Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let $H_1,\dots,...
- 26k
3
votes
0
answers
84
views
Low height integer points on a rank variety
Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition
$$
\mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1.
...
- 960
4
votes
0
answers
371
views
How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?
This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over $\...
- 22.8k
12
votes
0
answers
1k
views
Meaningful review of Moriwaki's "Arakelov Geometry"
I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry
book:
http://www.ams.org/bookstore-getitem/item=mmono-244
I could do the review the standard way in a day or ...
11
votes
1
answer
883
views
Higher Fano varieties and Tsen's theorem
The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
- 13.8k
1
vote
0
answers
351
views
Do those manifolds atrached to L-functions give rise naturally to motives? [closed]
Edited after Will Sawin's comment:
Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product $\...
- 6,984
0
votes
1
answer
598
views
Reference for a lemma on étale maps
The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ B\...
- 1,838
10
votes
1
answer
1k
views
how do automorphisms of elliptic curves act on the Tate module?
Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...
- 10.7k
4
votes
2
answers
411
views
Find all possible rational values of a parametric quartic such that it is reducible
Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 +...
- 441
11
votes
0
answers
173
views
Hyperelliptic curves over $\mathbb{Q}$ with a $\mu_p^2$ subgroup in their Jacobian
Given a prime number $p>2$, I'm looking for a smooth projective hyperelliptic curve $C$ defined over $\mathbb{Q}$ whose Jacobian $J(C)$ has a subgroup isomorphic to $\mu_p^2$ as a $Gal(\overline{\...
- 111
0
votes
0
answers
95
views
Rationality of intersection of algebraic groups
Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and $H(\...
user70017
30
votes
1
answer
2k
views
Enriques surfaces over $\mathbb Z$
Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...
- 149k
6
votes
0
answers
195
views
Non-embeddable varieties
Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$.
Then when ...
- 1,838
2
votes
1
answer
423
views
Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...
- 173
5
votes
1
answer
310
views
Twists of projective automorphisms
Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$
The twists of $X$ are classified by the Galois ...
- 21.3k
6
votes
2
answers
516
views
Obstruction and rational points on curves
Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
- 11.3k
0
votes
0
answers
166
views
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo $...
- 403
2
votes
1
answer
216
views
A question on polynomial heights
For a given polynomial $f(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$, define the height of $f$ as $H(f)$ as the maximum absolute value among its coefficients. We can also define the log ...
- 26.9k
0
votes
1
answer
263
views
Find all possible rational values of the parameter of a parametric cubic such that it is reducible
Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} \left({5\...
- 441
2
votes
1
answer
463
views
integral basis for the Lie algebra of the Neron model of an abelian variety
Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on $\mathcal{...
- 213
6
votes
0
answers
438
views
Brauer-Manin obstruction to surfaces of Kodaira dimension 1
Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
- 998
26
votes
4
answers
1k
views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
- 463
11
votes
0
answers
491
views
Can an abelian variety/Q have no non-trivial points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal (pro-)solvable ...
- 11.3k
10
votes
0
answers
340
views
Geometric vs combinatorial motives over Spec Z
Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
- 5,637
2
votes
0
answers
659
views
Constant group scheme and torsors
Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y \...
- 781
1
vote
1
answer
222
views
Automorphisms of $\mathbb F_q((\frac1T))$
I try to find the automorphisms $\sigma$ of $\mathbb F_q((\frac1T))$ with the following properties: $\sigma$ is an isometry,and $\sigma(\mathbb F_q[T])\subseteq\mathbb F_q[T]$. Almong the ...
- 3,998
19
votes
1
answer
3k
views
Mazur secret Bourbaki report "Analyse p-adique"
Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
3
votes
0
answers
134
views
ramified principal prime ideals in Artin-Schreier extension
Let $q$ be a power of $2$ and $K$ be the quadratic extension of $\mathbb F_q(T)$ defined by $K:=\mathbb F_q(T)[y]$ where $y^2+y=f(T)\in\mathbb F_q(T)$.
Put $\mathcal O_K$ the integral closure of $\...
- 3,998
0
votes
0
answers
261
views
Ring of integers in Artin-Schreier extension
Question put in mathstackexchange but received no answer.
It is well-known( see Goldschmidt book: Algebraic Functions and Projective Curves)
that for $q$ a power of $2$ a quadratic separable ...
- 3,998
150
votes
2
answers
22k
views
What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion ...
- 13.6k
2
votes
0
answers
194
views
Number of common solutions of polynomial system
Let $ \mathbb{F}_p$ be a finite field and $\{f_j\}_{j=1}^{j=r} \subseteq \mathbb{F}_p[X_1,...,X_n]$ be a set of polynomials.
Let consider the system of equations:
$f_j(x_1,...,x_n)=0$ for $j = 1,......
- 2,576
11
votes
3
answers
1k
views
References for general Hasse-Weil zeta function
Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...
- 17.6k
14
votes
1
answer
1k
views
Elliptic curves and connected components
Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.
- 345
14
votes
1
answer
2k
views
Some questions about the ring Z((x))
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
- 10.7k